modular-quantum-shor-compilation

name: modular-quantum-shor-compilation description: > Distributed compilation of Shor's algorithm on modular atomic quantum processors. Methodology for large-scale integer factorization across multiple quantum modules with optimized inter-module communication and intra-module clock rates. Use when: compiling Shor's algorithm for distributed quantum hardware, designing modular quantum architectures, optimizing quantum communication between modules, analyzing resource requirements for large-scale factoring, or planning fault-tolerant quantum cryptography attacks. Trigger words: Shor's algorithm, quantum factoring, modular quantum processor, distributed quantum compilation, RSA factoring, quantum cryptography, inter-module communication, Bell pairs, atomic processor.

Modular Quantum Shor Compilation

Overview

Methodology for compiling and optimizing Shor's algorithm across modular atomic quantum processors. Addresses the challenge of distributing ~10^6 physical qubits across multiple interconnected modules while minimizing the overhead from inter-module communication.

Based on: "Factoring 2048-bit RSA integers with a half-million-qubit modular atomic processor" (arXiv: 2605.03951, 2026-05-08)

Architecture

CPU-Inspired Modular Design

The processor architecture organizes quantum modules analogous to CPU cores:

• Modules: Each contains a subset of physical qubits with local operations
• Inter-module links: Bell pair distribution channels for remote operations
• Measurement units: Local measurement with specified latency (e.g., 1 ms)

Key Parameters

Compilation Strategy

Step 1: Problem Decomposition

Decompose the factoring problem into module-local and cross-module operations:

• Modular exponentiation: Core of Shor's algorithm, requires most gates
• Quantum Fourier Transform (QFT): Requires cross-module entanglement
• Measurement and classical post-processing: Determines factors from output

Step 2: Qubit Mapping

Map logical qubits to physical locations across modules:

• Data qubits: Distributed to minimize cross-module operations
• Ancilla qubits: Placed near frequently accessed data qubits
• Communication qubits: Dedicated qubits for Bell pair distribution

Step 3: Communication Optimization

Optimize the interplay between inter-module communication and intra-module clock rate:

• Pipelining: Overlap communication with local computation
• Batching: Group remote operations to amortize Bell pair setup cost
• Scheduling: Order operations to minimize idle time waiting for remote results

Step 4: Gate Compilation

Compile logical gates into module-local and cross-module primitives:

• Local gates: Direct execution within a module
• Remote CNOT: Teleportation-based using pre-distributed Bell pairs
• Measurement-based: Use measurement outcomes to control subsequent operations

Performance Analysis

Resource Scaling

For N-bit RSA integer factorization:

• Physical qubits: O(N^2) with surface code error correction
• Logical gates: O(N^3) for modular exponentiation
• Communication cost: Scales with the fraction of cross-module operations

Time Complexity

The distributed compilation achieves:

• 16% time overhead vs ideal single-module for 2048-bit RSA
• Linear scaling of overhead with communication latency
• Sub-linear scaling with number of modules (due to pipelining)

Practical Considerations

Error Correction

• Surface code or similar QEC required for fault tolerance
• Logical error rate must be below algorithm threshold
• Error correction overhead dominates physical qubit count

Communication Bottlenecks

• Bell pair distribution rate limits remote gate throughput
• Measurement latency affects feedback-dependent operations
• Network topology affects worst-case communication distance

Verification

• Classical verification of factoring result is O(N^2)
• Quantum volume benchmarks validate module performance
• Cross-module entanglement fidelity must exceed threshold

Pitfalls

• Underestimating communication overhead can negate parallelism benefits
• Module size must balance local computation vs communication frequency
• Error correction resource estimates vary significantly by code choice
• Classical preprocessing (selecting smoothness bounds) affects quantum resource needs